e expansion in cold atoms. Yusuke Nishida (Univ. of Tokyo & INT) in collaboration with D. T. Son (INT). Ref: Phys. Rev. Lett. 97, 050403 (2006). BCSBEC crossover and unitarity limit Formulation of e (=4d) expansion LO & NLO results Summary and outlook.
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e expansion in cold atoms
Yusuke Nishida (Univ. of Tokyo & INT)
in collaboration with D.T. Son (INT)
Ref: Phys. Rev. Lett. 97, 050403 (2006)
21COE WS “Strongly correlated manybody systems” 19/Jan/07
AttractionSuperconductivity/Superfluidity
BCS theory
(1957)
C.A.Regal and D.S.Jin,
Phys.Rev.Lett. 90, (2003)
Attraction is arbitrarily tunable by magnetic field
Swave scattering length :[0,]
Feshbach resonance
a (rBohr)
a>0
Bound state
formation
molecules
Strong coupling
a
a<0
No bound state
atoms
40K
Weak coupling
a0
Eagles (1969), Leggett (1980)
Nozières and SchmittRink (1985)
Superfluid phase

+
0
BCS state of atoms
weak attraction:akF0
BEC of molecules
weak repulsion:akF+0
Strong interaction : akF
Fermi gas in the strong coupling limit : akF=
Unitary Fermi gas
George Bertsch (1999),
“ManyBody X Challenge”
kF1
r0
V0(a)
Atomic gas : r0=10Å << kF1=100Å << a=1000Å
What are the ground state properties of
the manybody system composed of
spin1/2 fermions interacting via a zerorange,
infinite scattering length contact interaction?
0r0 << kF1<< a
kF is the only scale !
Energy per particle
x is independent of systems
cf.dilute neutron matter
aNN~18.5 fm >> r0~1.4 fm
Models
Simulations
Experiments
Duke(’03): 0.74(7), ENS(’03): 0.7(1), JILA(’03): 0.5(1),
Innsbruck(’04): 0.32(1), Duke(’05): 0.51(4), Rice(’06): 0.46(5).
Systematic expansion forx and other
observables (D,Tc,…) in terms ofe(=4d)
This talk
iT
iT
2component fermions
local 4Fermi interaction :
2body scattering at vacuum (m=0)
(p0,p)
=
n
1
Tmatrix at d=4e(e<<1)
Coupling with boson
g = (8p2e)1/2/m
is SMALL !!!
ig
ig
=
iD(p0,p)
Boson’s kinetic term is added,
and subtracted here.
=0 in dimensional regularization
Ground state at finite density is superfluid :
Expand with
Small coupling “g” between and
(g~e1/2)
Chemical potential
insertions (m~e)
k
p
p
=O(e)
+
p+k
k
p
p
p+k
=O(em)
+
“Counter vertices” to
cancel 1/e singularities
in boson selfenergies
1.
2.
O(e)
O(em)
or
or
Number of m insertions
Number of couplings “g ~e1/2”
+ O(e2)
Veff (0,m) =
+
+
O(e)
O(1)
Systematic
expansion !
pk
kp
iS(p) =
+
p
p
p
p
k
k
Aroundminimum
Expansion over 4d
Energy gap :
Location of min. :
0
NLO
corrections
are small
5 ~ 35 %
Good agreement with recent Monte Carlo data
J.Carlson and S.Reddy,
Phys.Rev.Lett.95, (2005)
cf. extrapolations from d=2+e
NLO are 100 %
Expansion around 4d
x
♦=0.42
2d boundary condition
2d
4d
d
Picture of weaklyinteracting fermionic &
bosonic quasiparticles for unitary Fermi gas may be a good starting point even at d=3
Z.Nussinov and S.Nussinov, condmat/0410597
2body wave function
Normalization at unitarity a
diverges at r0 for d4
Pair wave function is concentrated near its origin
Unitary Fermi gas for d4 is free “Bose” gas
At d2, any attractive potential leads to bound states
“a” corresponds to zero interaction
Unitary Fermi gas for d2 is free Fermi gas
iT
2component fermions
local 4Fermi interaction :
2body scattering in vacuum (m=0)
(p0,p)
=
n
1
Tmatrix at arbitrary spatial dimension d
“a”
Scattering amplitude has zeros at d=2,4,…
Noninteracting limits
iT
iT
Tmatrix at d=4e(e<<1)
Small coupling b/w fermionboson
g = (8p2e)1/2/m
ig
ig
=
iD(p0,p)
Tmatrix at d=2+e(e<<1)
Small coupling b/w fermionfermion
g = (2pe/m)1/2
ig2
=
g
g
d=4
Strong coupling
Unitary regime
BEC
BCS

+
d=2
Systematic expansions forx and other observables (D, Tc, …) in terms of “4d” or “d2”
Lagrangian
Power counting rule of
Arnold, Drut, and Son, condmat/0608477
x
NLO 4d
NLO 2d
d
NNLO 4d
Veff = + + + minsertions
NLO correctionis small ~4 %
Simulations :
Tc/eF
4d
2d
d
Critical exponents of O(n=1) 4 theory(e=4d1)
e expansion is
asymptotic series
but works well !
How about our case???